A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

1, 4, 6, 7, 10, 13, 16, 17, 20, 24, 25, 28, 30, 31, 35, 39, 41, 43, 45, 48, 51, 56, 57, 60, 62, 63, 65, 68, 71, 75, 79, 82, 83, 84, 87, 90, 94, 97, 98, 99, 102, 103, 105, 107, 110, 114, 117, 120, 122, 125, 127, 129, 133, 138, 141, 142, 144, 145, 148, 151, 152

Offset: 1

Author

Amiram Eldar, May 21 2023

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is (2-sqrt(2))/(3-sqrt(2)) = 0.369398... .

If A001694(k) is a term of A363190 then k and k+1 are consecutive integers in this sequence.

Examples

			The first 6 powerful numbers are 1, 4, 8, 9, 16 and 25. 1, 9 and 25 are odd and their positions in the sequence are 1, 4 and 6, respectively.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Teerapat Srichan, The odd/even dichotomy for the set of square-full numbers, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.
• Wikipedia, Powerful number.

Crossrefs

Cf. A001694, A062739, A363190, A363191, A363192.

Programs

Mathematica

Position[Select[Range[7000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(OddQ[#] &)] // Flatten

PARI

lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }